2,846 research outputs found
Solving Degenerate Sparse Polynomial Systems Faster
Consider a system F of n polynomial equations in n unknowns, over an
algebraically closed field of arbitrary characteristic. We present a fast
method to find a point in every irreducible component of the zero set Z of F.
Our techniques allow us to sharpen and lower prior complexity bounds for this
problem by fully taking into account the monomial term structure. As a
corollary of our development we also obtain new explicit formulae for the exact
number of isolated roots of F and the intersection multiplicity of the
positive-dimensional part of Z. Finally, we present a combinatorial
construction of non-degenerate polynomial systems, with specified monomial term
structure and maximally many isolated roots, which may be of independent
interest.Comment: This is the final journal version of math.AG/9702222 (``Toric
Generalized Characteristic Polynomials''). This final version is a major
revision with several new theorems, examples, and references. The prior
results are also significantly improve
Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems
We consider a class of Ising spin systems on a set \Lambda of sites. The
sites are grouped into units with the property that each site belongs to either
one or two units, and the total internal energy of the system is the sum of the
energies of the individual units, which in turn depend only on the number of up
spins in the unit. We show that under suitable conditions on these interactions
none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane,
where \beta is the inverse temperature and h the uniform magnetic field, touch
the positive real axis, at least for large values of \beta. In some cases one
obtains, in an appropriately taken \beta to infinity limit, a gas of hard
objects on a set \Lambda'; the fugacity for the limiting system is a rescaling
of z and the Lee-Yang zeros of the new partition function also avoid the
positive real axis. For certain forms of the energies of the individual units
the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the
negative real axis for all \beta. One zero-temperature limit of this type, for
example, is a monomer-dimer system; our results thus generalize, to finite
\beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of
monomer-dimer systems are real and negative.Comment: Plain TeX. Seventeen pages, five figures from .eps files. Version 2
corrects minor errors in version
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